[FOM] Technical Queries on SOL

[Joe Shipman]

Let X be the set of valid formulas in second order logic with standard (full) semantics. 

(1) what are upper and lower bounds for the logical strength of X?
(2) is there a “most inclusive” axiomatized deductive calculus to generate a fragment of X that includes all seriously proposed ones, or are there “competing” deductive calculi which are incompatible with each other?
(3) if the answer to (2) is the first alternative, what is the relationship of the fragment of X it generates to the set of formulas ZFC proves are valid for SOL with standard semantics?
(4) what is an example of a mathematical statement which we know that an oracle for X would not allow us to settle, of as low a type as possible?

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